An early goal from Aiden McGeady gave them the edge, but Celtic passed up a number of clear-cut chances before Stilian Petrov secured the victory with a stunning strike midway through the second half.
Celtic are now four points clear of Rangers at the top ahead of the Ibrox side's game against Motherwell.
Robert Douglas returned to the Celtic goal with Magnus Hedman out through illness and Hearts were without suspended duo Patrick Kisnorbo and Andy Webster.
Hearts took time to settle in defence and Chris Sutton came close after some slack play from Steven Pressley but his effort brushed the sidenetting.
The home side did not heed the warning and fell behind after nine minutes with a well-constructed goal.
Sutton combined with Jackie McNamara to set up Petrov who played the ball across the six-yard line for McGeady to apply the perfect finish.
There was a further setback for Hearts three minutes later when midfielder Michael Stewart had to go off with a groin injury and was replaced by Neil MacFarlane.
But the Tynecastle side stormed back and Douglas was well positioned to save a 20-yard free-kick from Jamie McAllister.
Celtic were always dangerous on the break and John Hartson should have made it 2-0 after some great work by Alan Thompson, but he sent his effort wide with Craig Gordon left exposed.
Mark de Vries had a chance at the other end, but Neil Lennon made a superb block to prevent the shot troubling Douglas.
Petrov wasted another chance for Celtic after Sutton took on Kevin McKenna but, like Hartson before him, the Bulgarian could not steer his shot on target.
Celtic had strong penalty claims six minutes from the break when Alan Maybury challenged Hartson in the area but referee Mike McCurry was not interested.
The second half followed a similar pattern with Celtic creating a number of opportunities but failing to put Gordon under any sustained pressure.
McGeady, Hartson and Petrov all had chances within minutes of the restart but all three failed to put their final shot on target.
Paul Hartley showed that Hearts were still in the contest when he gathered a knock-down from de Vries but pulled his shot wide.
De Vries then shot over from just a foot outside the penalty area as they sensed that they could yet cash in on Celtic's failure to take the chances that came their way.
But the points were taken out of their reach by a terrific strike from Petrov in the 68th minute. Hartson flicked the ball on for Sutton who played the ball on to his midfielder who crashed a shot into the top corner of the net.
Hartson was denied his place on the scoresheet in the dying minutes when he beat Gordon but Joe Hamill headed his effort off the line.